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      【IOI2018】排座位
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        <p><strong>题目链接：<a href="https://loj.ac/problem/2864" target="_blank" rel="noopener">2864. 排座位 - LibreOJ</a></strong></p>
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<p>首先说一句废话：对于一个确定的k，至多有一个大小为k的美妙矩形</p>
<p>那么我们考虑给定k如何判断是否存在大小为k的美妙矩形。我们设≤k的点为黑点，&gt;k的点为白点，如果所有黑点构成一个四连通块且连通块是一个矩形，那么这个连通块就是大小为k的美妙矩形。上述条件可以转化为以下两个条件：</p>
<ol>
<li>上面和左边均为白点的黑点恰有1个（保证黑点构成一个连通块）</li>
<li>上下左右至少有2个黑点的白点恰有0个（保证连通块是矩形）</li>
</ol>
<p>显然，满足条件1的黑点至少有1个，满足条件2的白点至少有0个，因此我们可以将满足条件1的黑点数和满足条件2的白点数加起来（这个数记作val(k)），然后判断val(k)是否为1</p>
<p>我们可以对每个k维护val(k)的值，不难想到线段树。对于矩形中一个确定的位置，它对val的贡献一定是一段连续的区间，所以修改一个位置时只需在线段树上做一下区间加法即可</p>
<p>对于一次交换，被交换的两个位置及其上下左右的格子对val的贡献都会受到影响，因此要对这10个位置都进行一次修改操作。那询问就简单了，判断val中的最小值是否为1，如果是1，返回最小值的数量即可</p>
<p>复杂度是1个log带上40倍常数，堪比2个log啊摔！</p>
<div class="highlight-box" autocomplete="off" autocorrect="off" autocapitalize="off" spellcheck="false" contenteditable="true" data-rel="CPP"><figure class="iseeu highlight /cpp"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span 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class="line">87</span><br><span class="line">88</span><br><span class="line">89</span><br><span class="line">90</span><br><span class="line">91</span><br><span class="line">92</span><br><span class="line">93</span><br></pre></td><td class="code"><pre><span class="line"><span class="meta">#<span class="meta-keyword">include</span><span class="meta-string">&lt;bits/stdc++.h&gt;</span></span></span><br><span class="line"><span class="meta">#<span class="meta-keyword">include</span> <span class="meta-string">"seats.h"</span></span></span><br><span class="line"><span class="keyword">using</span> <span class="keyword">namespace</span> <span class="built_in">std</span>;</span><br><span class="line"></span><br><span class="line"><span class="keyword">typedef</span> pair&lt;<span class="keyword">int</span>,<span class="keyword">int</span>&gt; pii;</span><br><span class="line"><span class="keyword">const</span> <span class="keyword">int</span> N=<span class="number">1000010</span>;</span><br><span class="line"><span class="keyword">int</span> mn[N&lt;&lt;<span class="number">2</span>],cmn[N&lt;&lt;<span class="number">2</span>],tag[N&lt;&lt;<span class="number">2</span>];</span><br><span class="line"><span class="keyword">int</span> *a[N],h,w,q,cnt;</span><br><span class="line"><span class="keyword">int</span> r[N],c[N];</span><br><span class="line">pii pos[<span class="number">10</span>];</span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">void</span> <span class="title">pushdown</span><span class="params">(<span class="keyword">int</span> o)</span></span></span><br><span class="line"><span class="function"></span>&#123;</span><br><span class="line">    <span class="keyword">if</span>(!tag[o]) <span class="keyword">return</span>;</span><br><span class="line">    tag[o&lt;&lt;<span class="number">1</span>]+=tag[o];</span><br><span class="line">    mn[o&lt;&lt;<span class="number">1</span>]+=tag[o];</span><br><span class="line">    tag[o&lt;&lt;<span class="number">1</span>|<span class="number">1</span>]+=tag[o];</span><br><span class="line">    mn[o&lt;&lt;<span class="number">1</span>|<span class="number">1</span>]+=tag[o];</span><br><span class="line">    tag[o]=<span class="number">0</span>;</span><br><span class="line">&#125;</span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">void</span> <span class="title">build</span><span class="params">(<span class="keyword">int</span> o,<span class="keyword">int</span> l,<span class="keyword">int</span> r)</span></span></span><br><span class="line"><span class="function"></span>&#123;</span><br><span class="line">    cmn[o]=r-l+<span class="number">1</span>;</span><br><span class="line">    <span class="keyword">if</span>(l==r) <span class="keyword">return</span>;</span><br><span class="line">    <span class="keyword">int</span> mid=(l+r)/<span class="number">2</span>;</span><br><span class="line">    build(o&lt;&lt;<span class="number">1</span>,l,mid);</span><br><span class="line">    build(o&lt;&lt;<span class="number">1</span>|<span class="number">1</span>,mid+<span class="number">1</span>,r);</span><br><span class="line">&#125;</span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">void</span> <span class="title">update</span><span class="params">(<span class="keyword">int</span> o,<span class="keyword">int</span> l,<span class="keyword">int</span> r,<span class="keyword">int</span> nl,<span class="keyword">int</span> nr,<span class="keyword">int</span> x)</span></span></span><br><span class="line"><span class="function"></span>&#123;</span><br><span class="line">    <span class="keyword">if</span>(l&gt;=nl&amp;&amp;r&lt;=nr)&#123;tag[o]+=x;mn[o]+=x;<span class="keyword">return</span>;&#125;</span><br><span class="line">    <span class="keyword">int</span> mid=(l+r)/<span class="number">2</span>;pushdown(o);</span><br><span class="line">    <span class="keyword">if</span>(nl&lt;=mid) update(o&lt;&lt;<span class="number">1</span>,l,mid,nl,nr,x);</span><br><span class="line">    <span class="keyword">if</span>(nr&gt;mid) update(o&lt;&lt;<span class="number">1</span>|<span class="number">1</span>,mid+<span class="number">1</span>,r,nl,nr,x);</span><br><span class="line">    mn[o]=min(mn[o&lt;&lt;<span class="number">1</span>],mn[o&lt;&lt;<span class="number">1</span>|<span class="number">1</span>]);cmn[o]=<span class="number">0</span>;</span><br><span class="line">    <span class="keyword">if</span>(mn[o&lt;&lt;<span class="number">1</span>]==mn[o]) cmn[o]+=cmn[o&lt;&lt;<span class="number">1</span>];</span><br><span class="line">    <span class="keyword">if</span>(mn[o&lt;&lt;<span class="number">1</span>|<span class="number">1</span>]==mn[o]) cmn[o]+=cmn[o&lt;&lt;<span class="number">1</span>|<span class="number">1</span>];</span><br><span class="line">&#125;</span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">void</span> <span class="title">gao</span><span class="params">(<span class="keyword">int</span> x,<span class="keyword">int</span> y,<span class="keyword">int</span> d)</span></span></span><br><span class="line"><span class="function"></span>&#123;</span><br><span class="line">    <span class="keyword">int</span> minv=h*w,val[<span class="number">4</span>],tot=<span class="number">0</span>;</span><br><span class="line">    <span class="keyword">if</span>(x&gt;<span class="number">0</span>) minv=min(minv,a[x<span class="number">-1</span>][y]);</span><br><span class="line">    <span class="keyword">if</span>(y&gt;<span class="number">0</span>) minv=min(minv,a[x][y<span class="number">-1</span>]);</span><br><span class="line">    <span class="keyword">if</span>(minv&gt;a[x][y]) update(<span class="number">1</span>,<span class="number">0</span>,h*w<span class="number">-1</span>,a[x][y],minv<span class="number">-1</span>,d);</span><br><span class="line">    <span class="keyword">if</span>(x&gt;<span class="number">0</span>) val[tot++]=a[x<span class="number">-1</span>][y];</span><br><span class="line">    <span class="keyword">if</span>(y&gt;<span class="number">0</span>) val[tot++]=a[x][y<span class="number">-1</span>];</span><br><span class="line">    <span class="keyword">if</span>(x&lt;h<span class="number">-1</span>) val[tot++]=a[x+<span class="number">1</span>][y];</span><br><span class="line">    <span class="keyword">if</span>(y&lt;w<span class="number">-1</span>) val[tot++]=a[x][y+<span class="number">1</span>];</span><br><span class="line">    <span class="keyword">if</span>(tot&lt;<span class="number">2</span>) <span class="keyword">return</span>;</span><br><span class="line">    sort(val,val+tot);</span><br><span class="line">    <span class="keyword">if</span>(val[<span class="number">1</span>]&lt;a[x][y]) update(<span class="number">1</span>,<span class="number">0</span>,h*w<span class="number">-1</span>,val[<span class="number">1</span>],a[x][y]<span class="number">-1</span>,d);</span><br><span class="line">&#125;</span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">void</span> <span class="title">addpos</span><span class="params">(<span class="keyword">int</span> x)</span></span></span><br><span class="line"><span class="function"></span>&#123;</span><br><span class="line">    pos[cnt++]=pii(r[x],c[x]);</span><br><span class="line">    <span class="keyword">if</span>(r[x]&gt;<span class="number">0</span>) pos[cnt++]=pii(r[x]<span class="number">-1</span>,c[x]);</span><br><span class="line">    <span class="keyword">if</span>(c[x]&gt;<span class="number">0</span>) pos[cnt++]=pii(r[x],c[x]<span class="number">-1</span>);</span><br><span class="line">    <span class="keyword">if</span>(r[x]&lt;h<span class="number">-1</span>) pos[cnt++]=pii(r[x]+<span class="number">1</span>,c[x]);</span><br><span class="line">    <span class="keyword">if</span>(c[x]&lt;w<span class="number">-1</span>) pos[cnt++]=pii(r[x],c[x]+<span class="number">1</span>);</span><br><span class="line">&#125;</span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">void</span> <span class="title">give_initial_chart</span><span class="params">(<span class="keyword">int</span> H,<span class="keyword">int</span> W,<span class="built_in">vector</span>&lt;<span class="keyword">int</span>&gt; R,<span class="built_in">vector</span>&lt;<span class="keyword">int</span>&gt; C)</span></span></span><br><span class="line"><span class="function"></span>&#123;</span><br><span class="line">    h=H;w=W;build(<span class="number">1</span>,<span class="number">0</span>,h*w<span class="number">-1</span>);</span><br><span class="line">    <span class="keyword">for</span>(<span class="keyword">int</span> i=<span class="number">0</span>;i&lt;h;i++) a[i]=<span class="keyword">new</span> <span class="keyword">int</span>[w];</span><br><span class="line">    <span class="keyword">for</span>(<span class="keyword">int</span> i=<span class="number">0</span>;i&lt;h*w;i++)</span><br><span class="line">    &#123;</span><br><span class="line">        r[i]=R[i];</span><br><span class="line">        c[i]=C[i];</span><br><span class="line">        a[r[i]][c[i]]=i;</span><br><span class="line">    &#125;</span><br><span class="line">    <span class="keyword">for</span>(<span class="keyword">int</span> i=<span class="number">0</span>;i&lt;h*w;i++) gao(r[i],c[i],<span class="number">1</span>);</span><br><span class="line">&#125;</span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">int</span> <span class="title">swap_seats</span><span class="params">(<span class="keyword">int</span> x,<span class="keyword">int</span> y)</span></span></span><br><span class="line"><span class="function"></span>&#123;</span><br><span class="line">    cnt=<span class="number">0</span>;</span><br><span class="line">    addpos(x);addpos(y);</span><br><span class="line">    sort(pos,pos+cnt);</span><br><span class="line">    cnt=unique(pos,pos+cnt)-pos;</span><br><span class="line">    <span class="keyword">for</span>(<span class="keyword">int</span> i=<span class="number">0</span>;i&lt;cnt;i++)</span><br><span class="line">        gao(pos[i].first,pos[i].second,<span class="number">-1</span>);</span><br><span class="line">    swap(a[r[x]][c[x]],a[r[y]][c[y]]);</span><br><span class="line">    swap(r[x],r[y]);</span><br><span class="line">    swap(c[x],c[y]);</span><br><span class="line">    <span class="keyword">for</span>(<span class="keyword">int</span> i=<span class="number">0</span>;i&lt;cnt;i++)</span><br><span class="line">        gao(pos[i].first,pos[i].second,<span class="number">1</span>);</span><br><span class="line">    <span class="keyword">return</span> (mn[<span class="number">1</span>]==<span class="number">1</span>)?cmn[<span class="number">1</span>]:<span class="number">0</span>;</span><br><span class="line">&#125;</span><br></pre></td></tr></table></figure></div>
      
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